Question

deriving the distance formula. use the explore tool to investigate the distance formula. to use the tool, press on the coordinates plane to move point c. explore the distance tool and then answer each question to determine the distance between points d and a. recall that the distance formula is √((x₂ - x₁)²+(y₂ - y₁)²). use the distance formula to determine the distance between points d and a. enter the square of the horizontal distance. (x₂ - x₁)² = . enter the square of the vertical distance. (y₂ - y₁)² = . enter the sum of the squares of the horizontal and vertical distances. (x₂ - x₁)²+(y₂ - y₁)² = . enter the approximate value of the sum. the distance between points d and a is approximately

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, for points $D$ and $A$, we first find $(x_2 - x_1)^2$ and $(y_2 - y_1)^2$.

Step2: Identify components

Let the coordinates of point $D$ be $(x_1,y_1)$ and of point $A$ be $(x_2,y_2)$. The horizontal - distance component is $(x_2 - x_1)^2$ and the vertical - distance component is $(y_2 - y_1)^2$.

Step3: Sum components

We sum $(x_2 - x_1)^2$ and $(y_2 - y_1)^2$ as per the formula: $(x_2 - x_1)^2+(y_2 - y_1)^2$.

Step4: Take square - root

Then we take the square - root of the sum to get the distance $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Answer:

The distance between points $D$ and $A$ is calculated using the formula $\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, where $(x_1,y_1)$ and $(x_2,y_2)$ are the coordinates of points $D$ and $A$ respectively. First find $(x_2 - x_1)^2$, then $(y_2 - y_1)^2$, sum them up and take the square - root of the sum.